The output of the household outlet is AC. But when repairing one of those, I found that there was a LED (used as indicator) in there. Since LED is a diode, it must switch on 60 times and switch off 60 times in a second, since frequency is 60Hz in my region. But I saw that the LED is constantly on. Is the LED really constantly on or is it my inability to see it switching on and off regularly?
Yes, such indicators often do flicker. Try sweeping your vision across it rapidly. You'll likely see a series of dots.
For example, here's an outlet in my kitchen:
And here's a picture of it taken with deliberate motion blur:
The flicker actually bothers me when I walk through the dark kitchen at night, in the same way that the PWM taillights on moderns cars do when driving at night.
The indicator you pictured looks more like a neon lamp than an LED to me.
But yes indicators do flicker based on the line frequency, for a neon or for a LED driven with a bridge rectifier the flicker will be at double line frequency (120Hz in north america 100Hz in Europe) which renders it invisible. OTOH if a LED is driven "half-wave" the flicker will be at line frequency (60Hz in north america, 50Hz in Europe) which is just on the edge of being noticable for humans.
Old neons also frequently develop a slow flicker (much slower than line frequency), i'm not an expert on why but there is some discussion at Why a neon lamp indicator on power strip switch flickers in the dark?
How can I confirm whether it is a neon lamp or LED?
There are a few clues.
The first is the visual appearance, Neons are glass bulbs with two identical looking wires inside and usually a lump sticking out the top where the bulb was sealed, LEDs are lumps of plastic, sometimes with a visible asymmetric electrode arrangement inside
The second is what it is connected to. Neons can be connected to the mains with a simple resistor. With LEDs you will generally find at least an diode in reverse parallel (to protect the LED from reverse voltage), a series capacitor (to drop the bulk of the voltage) and a series resistor (to protect against spikes).